Molecular dynamics simulation of thermal energy
transport in polydimethylsiloxane (PDMS)
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Luo, Tengfei et al. “Molecular Dynamics Simulation of Thermal
Energy Transport in Polydimethylsiloxane.” Journal of Applied
Physics 109.7 (2011): 074321. © 2011 American Institute of
Physics
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Molecular dynamics simulation of thermal energy transport in
polydimethylsiloxane
Tengfei Luo, Keivan Esfarjani, Junichiro Shiomi, Asegun Henry, and Gang Chen
Citation: J. Appl. Phys. 109, 074321 (2011); doi: 10.1063/1.3569862
View online: http://dx.doi.org/10.1063/1.3569862
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JOURNAL OF APPLIED PHYSICS 109, 074321 (2011)
Molecular dynamics simulation of thermal energy transport in
polydimethylsiloxane (PDMS)
Tengfei Luo, Keivan Esfarjani, Junichiro Shiomi, Asegun Henry, and Gang Chena)
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA
(Received 8 November 2010; accepted 27 February 2011; published online 12 April 2011)
Heat transfer across thermal interface materials is a critical issue for microelectronics thermal
management. Polydimethylsiloxane (PDMS), one of the most important components of thermal
interface materials presents a large barrier for heat flow due to its low thermal conductivity. In this
paper, we use molecular dynamics simulations to identify the upper limit of the PDMS thermal
conductivity by studying thermal transport in single PDMS chains with different lengths. We found
that even individual molecular chains had low thermal conductivities (j 7 W/mK), which is
attributed to the chain segment disordering. Studies on double chain and crystalline structures reveal
that the structure influences thermal transport due to inter-chain phonon scatterings and suppression
of acoustic phonon modes. We also simulated amorphous bulk PDMS to identify the lower bound of
PDMS thermal conductivity and found the low thermal conductivity (j 0.2 W/mK) is mainly due
C 2011 American
to the inefficient transport mechanism through extended vibration modes. V
Institute of Physics. [doi:10.1063/1.3569862]
I. INTRODUCTION
Thermal dissipation in microelectronic devices is an important issue which influences their performance. This issue
becomes more significant as the size of the device shrinks
and the power density increases. In a chip package, heat
must pass through several interface materials between the
die, the heat spreader and the heat sink. Among these interface materials, the polymer connecting the heat sink and the
heat spreader is one of the major barriers that impede thermal transport. Despite their low thermal conductivity, polymer based thermal interface materials (TIMs), also known as
thermal greases or thermal pastes, are necessary for filling
the surface voids and removing air gaps between the two
matching surfaces.1
Thermal grease is primarily made of silicone oil and
thickeners with different thermally conductive particles
added to increase its thermal conductivity. Polydimethylsiloxane (PDMS) is the most widely used silicone oil in industry. However, PDMS suffers from the common drawback
of all polymers -low thermal conductivity (0.15 W/mK)
(Ref. 2). Even with particle loading, PDMS-based thermal
greases have thermal conductivity ranging from 0.7–3 W/mK
with the highest being only about 5 W/mK with silver particles.3 Therefore, understanding thermal transport in PDMS
is critical to improve thermal interface materials.
Although it is known that PDMS has very low thermal
conductivity, detailed study of its thermal transport properties has been rare.4 It has been reported recently that single
polyethylene (PE) chain can have very high or even divergent thermal conductivity along the chain length direction,5,6
despite similar low thermal conductivity of its amorphous
counterpart to that of PDMS. It will be interesting to find out
a)
Author to whom correspondence should be addressed. Electronic mail:
gchen2@mit.edu.
0021-8979/2011/109(7)/074321/6/$30.00
whether the low thermal conductivity of bulk PDMS is due
to the amorphous structure disorder or the intrinsically low
thermal conductivity of the PDMS chains themselves. If
PDMS naturally has high thermal conductivity in the form of
straight chains, efforts can be made to engineer structures
favorable for thermal transport similar to the structural engineering accomplished with PE.7
In this work, we use nonequilibrium molecular dynamics (NEMD) simulation to systematically calculate the thermal conductivity of different PDMS structures from single
chain to amorphous bulk form. Different analyses are carried
out to study thermal transport mechanisms. The simulations
are carried out using large-scale atomic/molecular massively
parallel simulator (LAMMPS).8 Due to the presence of the
fast vibrating hydrogen atoms, a small time step of 0.25fs is
used. The condensed-phase optimized molecular potentials
for atomistic simulation studies (COMPASS) potential was
used, which has been optimized for PDMS by fitting to ab
initio calculations.9–11 Unlike some other polymer potentials
that use harmonic bond interactions, COMPASS uses anharmonic bonding terms, making it preferred for our thermal
transport study since anharmonicity is fundamentally linked
to umklapp phonon scatterings processes.
II. SINGLE PDMS CHAIN
Henry and Chen5,6 showed that a single PE chain can
have much higher thermal conductivity than its amorphous
counterpart, which sets an upper limit on the thermal transport. Similarly, the maximum thermal conductivity of PDMS
can be attained from the isolated single chain structure. To
construct the single chain simulation supercell, we start with
the basic segment of a PDMS molecule, which consists of
two silicon atoms, two oxygen atoms and four methyl (CH3)
groups [see Fig. 1(a)]. The optimized unit cell has a length
109, 074321-1
C 2011 American Institute of Physics
V
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074321-2
Luo et al.
FIG. 1. (Color online) Structure plot of (a) side-view and cross-sectional
view of a unit segment of PDMS, (b) a supercell of perfect and disordered
single-chain PDMS, and (c) an example NEMD setup and temperature profile with source temperature set to 325 K and sink temperature set to 285 K.
(O–red, Si–yellow, C–blue, H–white).
of about 5.0Å in the z-direction. The supercell for a singlechain PDMS is obtained by replicating the segment in the zdirection [see Fig. 1(b)].
To calculate thermal conductivity, we fixed the temperatures at the two ends of the sample using Langevin thermostats12 to establish a temperature gradient across the
simulation supercell [Fig. 1(c)]. Energies added into the heat
source and subtracted from the sink are recorded, and used
to calculate the heat flux. When steady state is attained, thermal conductivity is calculated using j ¼ ðJ=dT=dzÞ, where
j is thermal conductivity, J is heat flux and dT/dz is temperature gradient, which is obtained by fitting the linear part of
the temperature profile. It should be noted that for an isolated
single-chain, volume and cross-sectional area are not well
defined. We then use the effective volume of one PDMS
chain in the amorphous state as the volume of a single chain.
For example, an amorphous PDMS with a density of 0.92 g/
cc consisting of eight 25-segments chains has a total volume
3
of 5:37 104 Å . Thus a 25-segment single chain has an
3
effective volume of ð5:37 104 =8Þ ¼ 6:72 103 Å , yield2
ing an effective cross- sectional area of 27 Å . In NEMD
simulations, when subject to thermostats, such as Nose- Hoover and Langevin thermostats, the heat sink and source
regions are attached to fictitious heat reservoirs to maintain
their temperatures. These fictitious reservoirs can give raise
to artificial phonon distribution, which is different from the
intrinsic phonon distribution of the materials, in the sink and
source regions. Due to this mismatch of frequency spectra,
there can be distorted temperature profiles near the sink and
J. Appl. Phys. 109, 074321 (2011)
source region which lead to artificial boundary effects.
Depends on the nature of the boundaries, thermal transport
in the systems can be influenced to different extends.13 To
avoid these artifacts introduced by the thermostats, we
excluded the part of the temperature profiles near the sink
and source regions, and took only the linear part of the profile to obtain the temperature gradient for thermal conductivity calculation [Fig. 1(c)].
The unphysical phonon distribution from the sink and
source regions can sometimes affect a large distance from the
sink and source regions, if there is not enough phonon scattering to recover the correct distribution. This is especially a
problem for high thermal conductivity materials (e.g. Si),
which have long phonon mean free paths. However, for materials which have low thermal conductivity and strong phonon
scattering, this effect will be small. To gain confidence on our
NEMD setup, we did extensive tests and comparisons on a
200 segment with a chain length (L) of 100 nm by using
Nose-Hoover thermostats14,15 to fix the source and sink temperature (j ¼ 2.30 W/mK), the reversed-NEMD scheme16
(j ¼ 2.10 W/mK), fixed heat flux through velocity scaling
(j ¼ 2.21 W/mK), and by changing the sink and source region
lengths d (j ¼ 2.22, 2.44, 2.38 W/mK for d ¼ 3, 6, 9 nm,
respectively). The Langevin thermostat method yields a
j ¼ 2.36 W/mK. The variance in thermal conductivity among
different methods is less than 16%, which does not influence
the discussion of the results in the following contents.
We calculated thermal conductivities for system with
different chain lengths at 300 K, and the results are presented
in Fig. 2(a). During simulation, we found that the structure
of the perfect crystalline PDMS chain became disordered
easily by segment random rotation [see Fig. 1(b)]. For thermal transport, such disorder adds additional scattering to
thermal carriers besides the intrinsic umklapp scattering.
Due to the strong conformational disorder scattering, one
would not expect the thermal conductivity of the amorphous
PDMS chain to have large value. Our results show that single
chain PDMS keeps increasing even up to 1600 segments
(L ¼ 800 nm), though the thermal conductivity remains low
compared to PE.5,6,17 It is possible that some long wavelength modes are not significantly influenced by the local
conformational disorder and can still transport ballistically.
It can be shown, through Boltzmann transport equation and
the Matthiessen’s rule, that the inverse of the thermal conductivity (1/j) and the inverse of the sample length (1/L)
have linear relationship. When we plot 1/j against 1/L, we can
obtain the intrinsic thermal conductivity of an infinitely long
chain by linearly extrapolating the data to 1=L ¼ 0ðL ¼ 1Þ
[see Fig. 2(b)], and the thermal conductivity comes out to be
7.2 W/mK. As a comparison, we also used equilibrium molecular dynamics (EMD) with Green-Kubo method18 to calculate
the intrinsic thermal conductivity of single chains. In the EMD
simulations, periodic boundary conditions in the chain-length
directions are used, making the systems equivalent to chains
with infinite lengths, except for the size effect induced by the
cutoff phonon wavelength which equals to the simulation
supercell size. To study this size effect, we performed EMD on
supercells with different lengths and plotted the calculated thermal conductivity as a function of the cell length [see Fig. 2(c)].
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074321-3
Luo et al.
J. Appl. Phys. 109, 074321 (2011)
FIG. 3. (Color online) Single chain thermal conductivity for original
PDMS, “stiff” PDMS and disordered “stiff” PDMS.
FIG. 2. (Color online) (a) Thermal conductivity of single chain PDMS as a
function of the chain length from NEMD, (b) inverse thermal conductivity
of single chain PDMS plotted against the inverse chain length, showing a
linear dependence. Extrapolation of the line leads to the thermal conductivity of an infinitely long chain (dashed line). (c) Thermal conductivity of single chain PDMS as a function of the supercell length from EMD.
An ensemble averaging over 50 runs with different initial conditions were performed to obtain each thermal conductivity
datum. More detailed description of the EMD simulations can
be found in various references, such as Ref. 19. As we can
see, the calculated thermal conductivity converged to 6W/mK
with respect to supercell length. Thus, the intrinsic thermal
conductivities from both NEMD and EMD reasonably agree
with each other.
port and lead to a significant reduction in thermal conductivity. Such disorders, introduced by the random rotations of
the segments, lead to situations in which defining equilibrium atomic positions are not possible. In order to characterize the segment rotation, we calculated the backbone (Si-OSi-O) dihedral angle probability distribution of a single
PDMS chain (see Fig. 4). Very different from the sharp distribution profile of stiff polymers, like polyethylene oxide,20
the Si-O-Si-O dihedral angle has a wide-spread distribution.
Such a feature indicates the rotational softness of the PDMS
backbone about the chain axis which is the result of the very
flat torsional potential surface of the Si-O-Si-O dihedral
angle.10
To further characterize the role of disorder scattering in
PDMS thermal transport, we artificially increased the stiffness of the PDMS backbone by increasing the dihedral angle
energy constants so that there is no random segment rotation
during a molecular dynamic (MD) simulation. In this case,
the perfect lattice structure is maintained throughout the simulation. As can be seen from Fig. 4, the probability distribution of the “stiff” backbone dihedral angle has distinct sharp
peaks, suggesting that the temporal rotational movement of
the backbone is restricted. The calculated thermal conductivities of these “stiff” PDMS chains (Fig. 3) are much larger
than the original “soft” PDMS chains. We also performed a
simulation on a “stiff,” but “disordered” PDMS chain by
increasing the dihedral angle energy constants of an already
III. CONFORMATIONAL DISORDER SCATTERING
As we can see from Fig. 2(a), the thermal conductivities
of the conformationally disordered PDMS chains are small.
As a comparison, a single molecular chain of PE, which has
a very rigid conformation, has a much higher thermal conductivity (350 W/mK) (Ref. 6) than those of PDMS chains,
even though its phonon group velocity (16000 m/s) is only
about three times higher than that of PDMS (5600 m/s).
Thus, we believe that the conformational disorder scattering
in PDMS has a large impact on the intra-chain thermal trans-
FIG. 4. (Color online) Probability distribution of Si-O-Si-O dihedral angles
sampled from MD simulations.
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074321-4
Luo et al.
disordered PDMS chain. In such a system, the PDMS segments are randomly rotated, but no temporal rotation can
occur due to the increased dihedral angle energy constants.
The thermal conductivity of this disordered “stiff” chain
(cross in Fig. 3) is very close to that of the original PDMS
chain. This suggests that it is the conformational disorder,
instead of the temporal segment rotation, that leads to the
low thermal conductivity of the single PDMS chains.
IV. DOUBLE-CHAIN PDMS
We further simulated systems of PDMS with two chains
to calculate the thermal conductivity. We also observed
strong conformational disorder of the chain segments during
simulations. The calculated thermal conductivity as a function of supercell length is shown in Fig. 2(a) together with
the single chain thermal conductivity. Again, we used the
equivalent cross-sectional areas as described in the singlechain calculations, so that the thermal conductivities of the
single- and double-chain structures are directly comparable.
We found that the double-chain PDMS has lower thermal
conductivity and it appears to have converged within the
length scale considered. The major difference between single
chain and double chain PDMS is the interchain interaction,
which adds extra scattering sources for phonon transport
along the chains, and thus, can lead to a reduction in thermal
conductivity.
V. CRYSTALLINE PDMS
We also calculated the thermal conductivity of crystalline
PDMS by simulating 8-chain supercells (see Fig. 5). Chain
lengths with 50-, 100- and 200-segment long were simulated.
The supercells were prepared by iterative optimizations and
runs in NPT ensembles with initial arrangement illustrated in
Fig. 5(b). The cross-sectional areas were about 2 2 nm. As
can be seen from Figs. 5(a) and 5(c), after optimization, the
chain segments are disordered and closely packed. Once the
system is fully optimized in a compact pattern, there is no random rotation of chain segments as seen in single and double
chain structures. The calculated thermal conductivities [see
Fig. 2(a)], when converted using the equivalent cross-sectional area, are smaller than those of the single-chain and
J. Appl. Phys. 109, 074321 (2011)
double-chain PDMS, and they saturate when the chain length
exceeds 100 segments (50 nm). Compared to the double chain
structure, there are more surrounding chains for each chain,
thus, more scattering due to interchain interactions, which
lead to lower thermal conductivities.
VI. STRUCTURE INFLUENCE ON THERMAL
CONDUCTIVITY
The interchain interactions, working as extra forces on
chain segments, influence the motion of the chains. We
found that due to the presence of the additional chains, the
wavy motions of the backbone found in single chain PDMS
have lower amplitude in the double-chain structure and are
almost diminished in the crystalline PDMS. This observation
is characterized by the gyration radius (Rg) of the chain backbones, which is an indication of the amplitude of the chain
movement in the x-y plane. The gyration radius is defined as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
i
1X h
Rg ¼
mi ðxi xcm Þ2 þðyi ycm Þ2 ;
M i
where M is the total mass of a PDMS chain backbone, mi is
the mass of atom i in the backbone, and xcm, ycm are the
coordinates of the center of mass. The gyration radii of
the single-, double- and crystalline PMDS are 1:5548
6 0:0097Å; 1:527660:0137 Å and 1:434960:0054 Å,
respectively. This decreasing trend suggests the motions of
the chains in the x-y plane are confined when the structure
changes from single chain to double chain and to crystalline
PDMS. These wavy motions, corresponding to long wavelength, low frequency acoustic phonons, contribute a large
amount of thermal conductivity. Suppression of these modes
leads to the reduction of thermal conductivity. Such an effect
can be visualized by comparing the vibrational power spectrum from single chain, double chain and crystalline structures, which is calculated by taking the Fourier transform of
the velocity auto-correlation functions of the backbone
atoms (Si,O) (Fig. 6).
From Fig. 6, we can clearly see the decrease in the
power density of the acoustic phonons (< 20 cm1) when the
structures change from single-chain to double-chain, and the
power density of these modes is significantly suppressed
when PDMS is in a crystalline configuration, especially in
the low frequency spectrum. Such decreases in power density of acoustic modes are clearly associated with the change
of the structure. For a single PDMS chain, the chain can
vibrate freely due to its stand-alone structure. For the double-chain case, the vibration of each chain is restraint due to
the forces from another chain. In a closely packed crystalline
structure, each chain is surrounded by other chains which
confine its motion to a limited space, and the long wavelength motions found in single and double chains are
suppressed.
VII. AMORPHOUS PDMS
FIG. 5. (Color online) Crystalline PDMS: (a) cross-sectional view of an
optimized 8-chain supercell; (b) a schematic plot of the initial configuration
of the simulation box and chains; (c) side-view of the optimized supercell (8
chains are painted with different colors).
Finally, we simulated the amorphous bulk PDMS. The
modified Markov scheme21 is used to construct the amorphous supercell with a density of 0.92 g/cc. Periodic
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074321-5
Luo et al.
J. Appl. Phys. 109, 074321 (2011)
FIG. 8. (a) System size dependence and (b) temperature dependence of
thermal conductivity of amorphous bulk PDMS at a constant density.
FIG. 6. (Color online) Vibration power spectrum of single-chain, doublechain and crystalline PDMS (note that only spectra up to 120 cm1 is
shown).
boundary conditions are used in the transverse directions.
Heat sink and source are placed at the ends of the supercell
in the longitudinal direction (Fig. 7). The thermal conductivity is found to be around 0.2 W/mK at 300 K, which agrees
reasonably with the measured value of 0.15 W/mK.2
In an amorphous polymer, there are three means of thermal transport: (1) ballistic transport through long wavelength
propagating phonons, (2) diffusive transport through spatially extended nonpropagating vibration modes (extended
modes) and (3) energy hopping among localized modes due
to anharmonic transport.22 In amorphous materials, extended
modes are usually the most important carriers for thermal
transport. We also explore the roles of the other two means
of transport. If the long wavelength phonons, whose mean
free path are limited by boundary scatterings, dominate the
transport, thermal conductivity usually increases if the simulation supercell size increases in the thermal transport direction. We calculate the thermal conductivity of samples with
different lengths, and found that it is almost a constant
[see Fig. 8(a)], suggesting that the ballistic transport is not
important. The anharmonic energy transport among localized
modes depends on anharmonicity, a feature that increases
with temperature. We then calculated the temperature dependence of the amorphous PDMS thermal conductivity [see
Fig. 8(b)]. To exclude possible phase change effects on thermal conductivity, the same density is used at all temperatures. We found that temperature does not have a strong
influence on the thermal conductivity, suggesting that localized modes are not dominant thermal carriers.
VIII. CONCLUSION
FIG. 7. (Color online) Bulk amorphous PDMS and an example temperature
profile.
By using NEMD, we found that a single PDMS chain
has low thermal conductivity (6 W/mK at L ¼ 800 nm).
However, increase of thermal conductivity with chain length
is still observed with chain lengths of up to 800 nm because
long wavelength modes, which oversee the local disorder,
can still transport ballistically. Thermal conductivities from
MD of perfect “stiff” chains have much larger values compared to those from NEMD, suggesting that conformational
disorder in the chain segments greatly reduce the single
chain thermal conductivity. Double chain and crystalline
PDMS have lower thermal conductivity than single chain
structures due to interchain scattering and suppression of
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074321-6
Luo et al.
acoustic phonon modes. Amorphous PDMS has a thermal
conductivity of around 0.2 W/mK. Propagating modes and
localized modes are found to be insignificant to thermal
transport in amorphous systems. Overall, PDMS is found to
have low thermal conductivities even in a single chain form,
when compared to stiffer polymers such as PE. Our simulation results suggest that the low thermal conductivity is primarily due to conformational disorder within individual
chains and structure disorder among collections of chains.
ACKNOWLEDGMENTS
We would like to thank Dr. N. Yang, Dr. Y. Chalopin,
Dr. J. Jin and Ms. Z. Tian for valuable discussions. This
research was supported in part by National Science Foundation (Grant No. CBET-0755825), by DARPA NTI program
via Teledyne, and TeraGrid resources provided by TACC
Ranger (Grant No.TG-CTS100078).
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